SCHWEITZER 1994 Problem 3

Let p be an odd prime, A be a non-empty subset of residue classes modulo p, $f:A\to\mathbb R$ . Suppose that f is not constant and satisfies $f(x) \leq \frac{f(x + h) + f(x-h)}{2}$ whenever $x,x+h,x-h\in A$ . Prove that $|A| \leq \frac{p + 1}{2}$ .